det means 'it' or sometimes 'that'
'this' is det här or detta
'that' is det där or in some cases det
The best translation here is 'How is it possible?', but 'How is that possible?' is also a possible interpretation of the sentence, especially if you put a little more stress on det in the Swedish sentence.
No, when you use do in questions in English, you need another verb in the sentence. Like How do they do that? – in that case, the second do is the verb that carries the main meaning in the sentence. The first do is only there in order to make it a question. Or, say, What do you think? – 'do' makes it a question, and the question is about what someone thinks.
But in your sentence, there is no second verb.